the effect of inflation on inflation uncertainty in the g7 ...kushal banik chowdhury and nityananda...

Chowdhury and Sarkar-Eff. of Infl. on Infl. Uncer. in G7 Count.: A Double Threshold GARCH Model

34

The Effect of Inflation on Inflation Uncertainty in the G7 Countries:

A Double Threshold GARCH Model

Kushal Banik Chowdhury

and Nityananda Sarkar

Indian Statistical Institute and Indian Statistical Institute

ABSTRACT

This paper studies the impact of inflation on inflation uncertainty in a modelling

framework where both the conditional mean and conditional variance of inflation are

regime specific, and the GARCH model for inflation uncertainty is extended by including a lagged inflation term in each regime. Applying this model to the G7 countries with

monthly data from 1970 till 2013, it is found that the impact of inflation on inflation

uncertainty differs over the regimes in most of the G7 countries. The findings also provide strong empirical support to the well-known Friedman-Ball hypothesis of positive

impact of inflation on inflation uncertainty, but only for the high-inflation regime.

Key words: Inflation; Inflation Uncertainty; Regime Switching Model

JEL Classifications: C22; E31

1. INTRODUCTION

The impact of inflation on inflation uncertainty is well documented in theory. The empirical

literature on this topic is also sizeable. However, the findings on the nature of the relationship

and the empirical support for the link involving these variables vary with mixed results.

Consequently, the Friedman-Ball hypothesis (Friedman, 1977; and Ball, 1992), which states

that the effect of inflation on its uncertainty is positive, has found empirical support in varying

degrees1. While differences in the sample periods and frequencies of the data sets used could

account for mixed results, more importantly, it is the methodology or modelling approach

applied that would explain the varied findings.

It is worth mentioning that in many such empirical studies, a standard auxiliary assumption

typically made is that the parameters depicting the relationship are constant over the entire

sample period. This means that the causal links are assumed to be stable over time. However,

this assumption is far from innocuous and may often not hold in practice. Furthermore, there

is considerable evidence supporting the view that the economic time series are best thought of

as being generated by processes with time dependent parameters. According to several

studies, including Fischer (1993), Stock and Watson (1996), Caglayan and Filiztekin (2003),

Arghyrou et al. (2005) and Lanne (2006), social, economic and political changes are likely to

Kushal Banik Chowdhury, Economic Research Unit, Indian Statistical Unit, 203 B. T. Road, Kolkata-700108, India, (email: [email protected]), Tel: +919051906848.

Nityananda Sarkar, Indian Statistical Institute, 203 B.T. Road, Kolkata-700108, India, (email: tanya@isical

.ac.in). 1 See, for details on the evidence of positive effect of inflation on inflation uncertainty, Grier and Perry (1998),

Fountas (2001), Fountas et al. (2002), Kontonikas (2004), Daal et al. (2005), and Fountas and Karanasos (2007),

and for negative effect, Engle (1983), Bollerslev (1986), Cosimano and Jansen (1988), Hwang (2001), and

Wilson (2006).

mailto:[email protected]:[email protected]:[email protected]

International Econometric Review (IER)

35

make the macroeconomic relationship with constant parameter linear models quite untenable,

especially when the length of the time series is long enough.

In fact, a recent trend in the literature on the relationship between inflation and inflation

uncertainty is that this relationship is assumed to be neither linear nor stable over time. While

there are a number of modelling procedures that incorporate these important aspects, an

important class of such models is the regime switching models. In case of inflation, regime

switching models are quite relevant and appropriate since inflation, by its very nature as a

macro variable, may change for a period of time before reverting back to its original

behaviour or switching to yet another style of behaviour, often due to intervention by

government and/ or regularity authorities. The issue of regime switching behaviour of

inflation while dealing with inflation uncertainty, was first raised by Evans and Wachtel

(1993). They concluded that adequate attention needs to be paid to the consequences of

regime switches; otherwise, existing models without any consideration to regime changes

would seriously underestimate both the degree of uncertainty of inflation and its impact.

Following their lead, some researchers including Kim (1993), Kim and Nelson (1999), Bhar

and Hamori (2004), Chang and He (2010), and Chang (2012) have studied the relationship

between inflation and its uncertainty where the regime specific behaviour have been duly

incorporated in the model. A point worth noting is that all the above-mentioned studies have

used the concept of unobserved regime and accordingly applied the well-known Markov

switching regression model2.

There is another class of regime switching models where regimes can be characterized by a

threshold variable that is observable (see, Tong and Lim, 1980; Chan and Tong, 1986;

Granger and Teräsvirta, 1993; Teräsvirta, 1998; Li and Li, 1996; and Brooks, 2001; for details

on such models). This work, which essentially studies the impact of inflation on inflation

uncertainty, takes be the past level of inflation as a threshold variable. The notion of taking

the level of inflation as the observed threshold variable has been influenced by a few recent

studies. For instance, Baillie et al. (1996) applied an autoregressive fractionally integrated

moving average (ARFIMA) model to describe the long memory process of inflation dynamics

for ten countries. They found that for six low inflation countries viz., Canada, France,

Germany, Italy, Japan, and the USA, there is no apparent relationship between mean and

variance of inflation. However, for the high inflation economics of Argentina, Brazil, Israel,

and the UK, there is strong support for the Friedman hypothesis. In a recent study, Chen et al.

(2008) have observed that the effect of inflation on inflation uncertainty is asymmetric. To be

more specific, they have found a U-shaped pattern for four countries of East Asia, based on a

nonlinear flexible regression model of Hamilton (2001), suggesting that inflation uncertainty

is more sensitive to inflation in an inflationary period than in a deflationary period. Thus, it is

relevant as well as important to examine the dynamic interaction between inflation and

inflation uncertainty in the different regimes considered, and also to find if these interactions

are different across the different regimes.

The purpose of this study is to examine the effect of inflation on inflation uncertainty in a

regime switching modelling framework where regimes are determined by the past level of

inflation. This model has two other additional features as well. The first is that not only the

conditional mean model but also the conditional variance model is regime specific. Since

(G)ARCH (Engle, 1982; and Bollerslev, 1986) or similar other models are preferred when

2 In this class of Markov switching regression models, regimes are not deterministic and can be determined by an

underlying unobservable stochastic process depending upon the probabilities assigned to the occurrence of the

different regimes.


36

measuring inflation uncertainty, we too have applied the GARCH model; accordingly, the

proposed model is a double-threshold GARCH (DTGARCH) model, originally proposed by

Li and Li (1996) in the context of stock returns3. The second feature for our model involves

the extension of the usual GARCH specification by explicitly incorporating a lagged inflation

term in the conditional variance specification, the coefficient of which depicts the impact of

inflation on inflation uncertainty. One distinct advantage of this model is that it allows for

different behaviours of inflation uncertainty in different regimes, including the one captured

through the lagged inflation term. We have applied this model to the time series covering the

period from January 1970 to December 2013 for all members of the G7 countries. In this

context it may be mentioned that since we have considered a long enough time period, we

have specifically considered the issue of structural break in our study. However, unlike others,

we have dealt with the twin issues of structural break and stationarity together by applying the

recently developed tests of Perron and Yabu (2009) and Kim and Perron (2009).

The format of the paper is as follows: Section 2 describes the proposed model. Section 3

discusses empirical findings. Section 4 ends the paper with some concluding observations.

2. THE PROPOSED MODEL

Fountas et al. (2000) first considered the model for inflation uncertainty to explicitly include a

lag inflation term. Like most studies, they used the GARCH (1,1)4 model as the basic model

to measure inflation uncertainty and then augmented it by including the first lag of inflation.

The conditional mean model was taken to be the usual AR model. Thus, their model,

designated as the AR(k)-GARCH(1,1)L(1)5, consists of the following specifications for the

conditional mean and conditional variance.

πt =

ϕ0

+

ϕ1

πt-1

+

ϕ2

πt-2

+

⋯ + ϕk πt-k

+

εt, εt|Ψt-1

~

N(0,

hπ,t) (2.1)

hπ,t =

ω

+

αε

2t-1

+

βht-1

+

θπt-1 (2.2)

where πt stands for the value of stationary inflation at time t, Ψt-1 ={πt-1,

πt-2,…} is the

information set at t -

1, hπ,t is the conditional variance of πt at t, and k the optimal lag order in

the conditional mean specification. All roots of the polynomial (1 -

ϕ1B

-

⋯ - ϕk B

k)

=

0 are

assumed to lie outside the unit circle for stationarity of πt, and the usual restrictions on the

parameter in hπ,t are assumed to ensure positivity. Here the coefficient θ depicts the impact of

inflation on inflation uncertainty. Obviously, a significant positive value of θ provides

empirical support to the Friedman-Ball hypothesis for a given time series on inflation. We use

this model as a benchmark model to find if the introduction of regimes in both inflation and

inflation uncertainty leads to better understanding and modelling involving these two

variables.

In describing the proposed model, which is a regime-dependent model with two regimes, we

allow the effect of inflation on its uncertainty to be different in the two regimes. As already

3 See also, Chen (1998), Brooks (2001), Chen et al. (2003), Chen and So (2006), Chen et al. (2006), and Yang

and Chang (2008) for applications of the DTGARCH model for other financial variables. 4 For our study, the conditional mean model has been taken to be the AR model on the basis of behaviour of the

autocorrelation function (ACF) and partial autocorrelation function (PACF) of the inflation series. The value of k

has been determined by the Akaike’s information criterion (AIC). As regards the values of p and q of the

GARCH(p,q) specification for conditional variance, p = q = 1 has been found to be adequate, and hence

GARCH(1,1) model has been considered. 5 ‘L(1)’ stands for the fact that the specification for hπ,t includes the first lag of inflation as a separate term.


37

stated in the preceding section, we implicitly assume that the threshold variable which defines

the regime that occurs at any given point in time, is known, and takes the preceding value of

inflation i.e., πt-1, to be the threshold variable. We define the two regimes according to the

value of stationary inflation at t - 1, i.e., πt-1

>

0 and πt-1

≤

0, which are being labelled as high

and low inflation regimes6, respectively.

The proposed model is a generalization of the model in Fountas et al. (2000) in that here

regimes are introduced in the specifications of both the conditional mean and conditional

variance of inflation. Obviously, the proposed model is an extension of the DTGARCH

model, where the conditional variance specification includes a lag inflation term. The

resultant model, denoted as DTGARCH(1,1)L(1), is thus specified as follows:

πt =

(ϕ

l0

+

ϕ

l1πt-1

+

ϕ

l2πt-2

+

⋯ + ϕlkπt-k)

I[πt-1

≤

0]

+

(2.3)

+ (ϕ

h0

+

ϕ

h1πt-1

+

ϕ

h2πt-2

+

⋯ + ϕhkπt-k)

(1

-

I[πt-1

≤

0])

+

εt, εt|Ψt-1

~

N(0,

hπ,t)

hπ,t =

(ω

l +

α

l ε2t-1

+

β

l ht-1

+

θ

l πt-1)

I[πt-1

≤

0]

+

(2.4)

+ (ω

h +

α

h ε

2t-1

+

β

h ht-1

+

θ

h πt-1)

(1

-

I[πt-1

≤

0])

where I[.] is the indicator function which takes the value 1 if πt-1 ≤

0 and 0 otherwise, and

Ψt-1 =

{πt-1,

πt-2,…} is the information set at t

-

1. In this model coefficient vector

ξl =

(ϕ

l0,

ϕ

l1,...,

ϕ

lk,

ω

l , αl , β

l , θl )' comprises the coefficients in both conditional mean and

conditional variance specifications for the low inflation regime and similarly for the high

inflation regime ξh =

(ϕ

h0,

ϕ

h1,...,

ϕ

hk,

ω

h , αh , β

h , θh )'. Apart from the usual GARCH coefficients, θ

l

and θh denote the effects of inflation on inflation uncertainty in the low and high inflation

regimes, respectively.

The parameters of each of the benchmark and the proposed models i.e., DTGARCH(1,1)L(1)

and AR(k)-GARCH(1,1)L(1) models have been estimated by the maximum likelihood (ML)

method of estimation under the assumption of normality7 for the conditional distribution of εt.

For the iterative optimization procedure involved in the estimation process, the well-known

Berndt-Hall-Hall-Hausman (BHHH; Berndt et al., 1974) algorithm has been used. The

necessary computations have been done in GAUSS.

3. EMPIRICAL ANALYSIS

Like most of the empirical studies on the relationship between inflation and inflation

uncertainty (see, for instance, Grier and Perry, 1998; Fountas et al., 2002; Fountas et al.,

2004; Bredin and Fountas, 2005), we have used monthly data for this study. We have taken

monthly time series of consumer price index (CPI) as a measure of the monthly price level for

each of the G7 countries viz., Canada, France, Germany, Italy, Japan, the UK, and the USA.

These time series on CPI have been downloaded from the official website of Federal Reserve

Bank of St. Louis (http://research.stlouisfed.org/). Since the available time series on CPI is

not adjusted for seasonality, each series has been adjusted for seasonality by applying the

X12-ARIMA filtering method. The time period considered is from January 1970 to June

6 The reasons for this labelling are stated in the next section (p. 41). 7 It may be noted that like many financial time series, inflation has been found to be non-normal. However,

assumption of normal distribution for estimation is very common since the resulting estimates often converge

unlike some non-normal error distributions. In this particular literature on modelling inflation, the assumption of

normal distribution for the error is most common, and hence we have also made this assumption of normal

distribution for our analysis.

http://research.stlouisfed.org/


38

2013. The total number of observations is 522. The time series on inflation, π t, has been

obtained as the differences of the logarithm values of the CPI in percentage i.e.,

π t =log(CPIt/CPIt-1)*100

The time series of inflation for all the G7 countries are plotted in Figures 3.1. It is visually

evident that all the inflation series follow a trend - although segmented in nature, especially in

case of Canada, France, Italy, the UK, and the USA. Further, it appears from the plots that

there may be at least one structural break in each of the seven time series on inflation.

Figure 3.1 Time series plots on inflation of the G7 countries

-1

0

1

2

3

1970 1975 1980 1985 1990 1995 2000 2005 2010

Canada

-0.5

0.0

0.5

1.0

1.5

2.0

1970 1975 1980 1985 1990 1995 2000 2005 2010

France

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

1970 1975 1980 1985 1990 1995 2000 2005 2010

Germany

-0.4

0.0

0.4

0.8

1.2

1.6

2.0

2.4

2.8

1970 1975 1980 1985 1990 1995 2000 2005 2010

Italy

-1

0

1

2

3

4

1970 1975 1980 1985 1990 1995 2000 2005 2010

Japan

-1

0

1

2

3

4

5

1970 1975 1980 1985 1990 1995 2000 2005 2010

The UK

-2

-1

0

1

2

1970 1975 1980 1985 1990 1995 2000 2005 2010

The USA

We present the summary statistics on inflation, covering the sample period, for all the seven

countries in Table 3.1 below.

Country Canada France Germany Italy Japan The UK The USA

Mean 0.346 0.372 0.233 0.555 0.218 0.472 0.349

Std. dev. 0.379 0.359 0.251 0.508 0.498 0.513 0.331

Skewness 0.613 0.849 1.172 1.572 2.620 2.514 0.329

Kurtosis 4.765 3.245 7.066 5.594 14.544 14.298 6.650

Jarque-Bera 100.33* 63.92* 478.22* 360.62* 3489.17* 3319.76* 298.62*

Table 3.1 Descriptive statistics on inflation.

Notes: * indicates significance at 1% level of significance.

From this table it is evident that among the seven countries, Italy exhibits the highest average

inflation of 0.555 per cent with the standard deviation of 0.508 per cent. The three countries

viz., Italy, Japan and the UK have larger standard deviations than the remaining four

countries. The skewness and kurtosis values indicate that all the seven inflation series are

positively skewed and have leptokurtic distributions. Worthwhile to highlight are the high

values of skewness for Japan and the UK at 2.620 and 2.514, respectively. As for the kurtosis,

Japan again has the highest value of 14.544, followed by the UK with a value of 14.298. The

kurtosis values for Germany and Italy are also moderately high. Therefore, as expected,

results of the Jarque-Bera normality test clearly indicate that the null hypothesis of normality

is rejected for all the inflation series, in particular, with very high test statistic value for

countries like Japan and the UK.


39

We now discuss the stationarity/non-stationarity status of the inflation series. Since structural

change in a time series affects inference on unit roots of the series, we examined these two

issues together following the recently-developed procedures of Perron and Yabu (2009) and

Kim and Perron (2009).

3.1. Stationarity and Structural Break

Since 1960s most of the industrialized countries including the G7 have experienced long-term

swings in levels of inflation. Inflation had progressively risen in the 1960s and 1970s before it

declined in the 1980s. Inflation further declined in the early to mid-1990s and since then

remained low and stable (see, for details, Blinder, 1982; DeLong, 1997; Clarida et al., 2000;

Orphanides, 2003; Meltzer, 2005; and Nelson, 2005). These observations have led many

researchers to analyse the statistical properties of inflation persistence over the last two

decades. This statistical issue has special relevance for inflation since in the earlier studies on

developed economies, especially those on the UK and the USA, the unit root status of the

time series were found to be mixed (see Kontonikas, 2004; Bhar and Hamori, 2004; Fountas

et al., 2004; Daal et al., 2005; Conrad and Karanasos, 2006; and Fountas et al., 2006; for

details). Thus, in this context, the span of the data sets used in this study viz.,

January 1970

-

June

2013, is quite large. Hence, structural breaks in the trend function of these

series are likely to occur, and as Perron (1989) first pointed out, disregarding this could

mislead the conclusion on the presence of unit roots in the series.

To test for stationarity of a time series, most often the augmented Dicky-Fuller (ADF; Dicky

and Fuller, 1979) test is used, where the null hypothesis of non-stationarity is tested against

the alternative of stationarity. However, due to the influential work by Perron (1989), the

commonly used ADF test has been criticized because of its bias towards non-rejection of the

null hypothesis of a unit root against the alternative of trend stationarity in the presence of a

structural break in the deterministic trend, as well as for its low power for near integrated

process. Subsequently, Perron (1989, 1990) proposed an alternative unit root test that allows

for the possibility of a structural break in the trend function under both the null and alternative

hypotheses. However, one serious limitation of this test is that it is based on the assumption of

a known break date. Subsequently, Zivot and Andrews (1992), Perron (1997), and Vogelsang

and Perron (1998), among others, have treated the break date to be unknown, which is

endogenously determined from the model.

However, recently Kim and Perron (2009) have pointed out that in all these tests with an

endogenous break point i.e., those by Zivot and Andrews (1992), Perron (1997), and

Vogelsang and Perron (1998), a trend break is not allowed under the null hypothesis. These

tests consider a break in the time series under the alternative only. Hence, tests of this kind are

likely to be affected in terms of size and power. To overcome this limitation, Kim and Perron

(2009) have developed a new test procedure on the line of Perron’s (1989) origina l

formulation of trend break being allowed under both the null and alternative hypotheses, but

the break date is now assumed to be unknown.

Now, prior to applying this unit root test, knowing whether a structural break is present in a

given time series is crucial, especially when the break date is assumed to be unknown, as in

the case with the Kim-Perron (2009) test. Further, in this context, in the absence of a trend

break, the well-known ADF test has the highest power compared to any other alternative test,

and it is the most appropriate test for testing the presence of unit roots in a time series. It is,

therefore, all the more important that the knowledge of the presence or absence of a break in


40

the trend function is available before deciding on the appropriate unit root test. However, as

pointed out by Perron and Yabu (2009), testing for a structural break in the trend function

depends on whether the noise component is stationary or non-stationary with unit roots. Thus

there is some sort of a circular problem in testing for these twin issues. To deal with this,

recently Perron and Yabu (2009) proposed a test for structural change in the trend function of

a univariate time series, which can be performed without any prior knowledge on whether the

noise component is stationary or non-stationary containing unit roots8. Since this test is very

general in its approach insofar as the assumption on noise is concerned, we have performed

this test to detect the presence of structural break, if any, in the trend function of a time series.

Thus, in case the Perron-Yabu test suggests that there is no break in the deterministic trend,

we apply the usual ADF test; otherwise, we use the Kim-Perron test to test the null hypothesis

of unit roots against the alternative of stationarity with a break in the deterministic trend

function under both hypotheses9.

Three models - Model I, Model II and Model III - representing a single change in intercept

only, a one-time change in the slope of linear trend function without a change in level, and a

simultaneous change in the intercept and the slope coefficients, respectively, are considered

for the Perron-Yabu test. For our study, we have considered Model III only. The test statistic

values with trimming percentage being 0.15 are reported in Table 3.2.

Country

Perron-Yabu

structural break test

Kim-Perron

unit root test

Estimated break date

Canada 53.029* -21.615* June 1982

France 48.350* -8.000* September 1983

Germany 8.741* -12.948* October 1982

Italy 12.557* -5.034* December 1982 Japan 21.985* -12.279* December 1976

The UK 41.028* -7.767* December 1981

The USA 27.492* -15.488* September 1981

Table 3.2 Results of tests for structural break and unit roots in inflation series.

Notes: *, ** and *** indicate significance at 1%, 5% and 10% levels of significance, respectively.

The Perron-Yabu test statistic values are significant for all seven inflation series, which

clearly establish that either or both of the intercept and slope of the trend function have

undergone structural change. Thus the test confirms the presence of one significant change in

the deterministic trend of inflation for each of the G7 countries.

With this empirical finding on the presence of a structural break in the trend function of

inflation, we have applied the third variant of additive outlier model10

, Model A3 in Kim and

Perron (2009), to each of the seven inflation series to test the null hypothesis of unit roots

with a deterministic trend break against the alternative of stationarity with a break in the

deterministic trend. The values of this test statistic are reported in the 3rd

column of Table 3.2.

The computed values of the t-ratio are compared with the appropriate critical values available

8 Perron and Yabu (2009) have considered a quasi-feasible generalized least squares technique that uses a super-

efficient estimate of the sum of autoregressive parameters, which governs the stationary or integrated behaviour

of a time series. 9 A GAUSS program code for Perron and Yabu (2009) structural break test and a MATLAB code for Kim and

Perron (2009) unit root test in presence of a structural break have been taken from the official website of Pierre

Perron. 10 Kim and Perron (2009) have considered three types of additive outlier models viz., A1, A2 and A3,

representing the occurrence of a structural break in the intercept only, in the slope only, and both in the intercept

and slope coefficients of the trend function, respectively.


41

in Perron (1989) and Perron and Vogelsang (1993). The findings clearly suggest that the

inflation series of each of these countries are stationary when accounting for the presence of

one structural break. In each case, the null hypothesis of a unit root with a trend break against

the alternative of a stationary process with a trend break is rejected. We thus conclude that the

underlying data generating process for each of the seven inflation series is a trend stationary

process (TSP), having a structural break in the respective deterministic trend functions.

The stationary time series on inflation for each country, denoted by π t, has then been obtained

after removing the segmented deterministic trend from π t. Specifically, this has been obtained

by regressing π t on an intercept, an intercept dummy, a linear trend term and a trend (linear)

dummy, and then collecting the detrended series, π t, which is now stationary in the sense of

having neither any stochastic trend or any deterministic trend. Thus the obtained stationary

time series on inflation has been used for all subsequent analyses. In this context it may be

noted that the two inflation regimes, defined as π t-1 ≤

0 and π t-1

>

0 in Section 2, essentially

refer to π t-1 being less than or equal to and greater than some positive values, as indicated by

their respective deterministic trend function of each inflation series. Hence the two regimes

are referred to as the low-inflation and high-inflation regimes, respectively.

The estimated break dates are reported in the 4th

column of Table 3.2. The estimated break

dates refer to the period of 1970s and 1980s. Thus these findings provide empirical support to

the general observation that the high phase of inflation during the 1960s-1970s started

reducing substantially towards stabilization in the 1980s. There are quite a few studies

supporting this observation as well. For instance, in their study, Cecchetti and Krause (2001)

have reported that inflation in most developed and developing countries remained at a very

high level during the period of ‘great inflation’ in 1960s, which was coupled with the oil price

shock in 1973. The volatility appears to be more pronounced during that period while

remaining low and stable during the latter half of 1980s due to marked improvement in

macroeconomic performances in those countries. This finding has also been supported by

Stock and Watson (2002), who pointed out that during the period of 1990s, not only volatility

of inflation decreased sharply but also average inflation underwent a downward trend.

Further, Krause (2003) reported that in a cross-section of 63 countries, mean inflation has

fallen from approximately 83 per cent in the pre-1990 period to approximately 9 per cent in

the latter half of 1990s. Given such empirical findings, it is expected that there would be at

least one structural break in the time series on inflation, and this is, in fact, what we have

found11

.

3.2. Empirical Findings on The Impact of Inflation on Inflation Uncertainty

In this section we first present the estimates of the benchmark AR(k)-GARCH(1,1)L(1)

model. Next we discuss the results of estimation and testing of hypotheses of interests

involving the parameters of the proposed model.

3.2.1. The Benchmark AR(k)-GARCH(1,1)L(1) Model

The results of estimation of the benchmark AR(k)-GARCH(1,1)L(1) model specified in

equations (2.1) and (2.2) are reported in Table 3.3. As noted in the preceding section, the

11 Clark (2006) and Levin and Piger (2003) allowed the possibility of structural breaks while examining the

persistence of inflation. Rapach and Wohar (2005) also tested for breaks and found evidence of shift in the level

of inflation in a wide range of European countries.


42

optimal lag orders of the autoregressive terms for the seven inflation series have been

obtained by AIC.

Canada France Germany Italy Japan The UK The USA

Conditional mean

ϕ0 0.007 0.006 0.003 -0.004 0.003 -0.012 -0.004

ϕ1 -0.012 0.305 * 0.078 0.266 * 0.082 *** 0.208 * 0.351 *

ϕ2 0.141 * 0.026 0.184 * 0.199 * 0.037 0.298 * 0.016

ϕ3 0.094 ** 0.114 ** 0.110 ** 0.150 * 0.021 0.124 ** -0.008

ϕ4 0.049 0.020 0.058 0.032 0.043 -0.096 ** 0.099 **

ϕ5 0.113 ** -0.018 0.015 -0.016 0.114 ** 0.057 -0.029

ϕ6 0.016 0.052 0.117 * 0.119 ** 0.029 0.114 * 0.019

ϕ7 0.035 0.080 *** 0.038 0.091 *** 0.082 *** - 0.067 ***

ϕ8 - 0.045 0.067 *** 0.032 0.025 - -0.058

ϕ9 - -0.003 - 0.027 0.150 * - 0.062

ϕ10 - 0.057 *** - 0.026 0.044 - 0.055

ϕ11 - - - -0.046 0.051 - -

ϕ12 - - - -0.125 * -0.133 * - -

Conditional variance

ω 0.011 * 0.009 ** 0.013 * 0.001 *** 0.002 *** 0.008 * 0.004 *

α 0.237 * 0.181 * 0.464 * 0.160 * 0.080 ** 0.675 * 0.271 *

β 0.662 * 0.553 * 0.347 * 0.830 * 0.896 * 0.437 * 0.692 *

θ 0.025 ** 0.007 0.023 0.002 -0.007 -0.007 0.024 *

MLLV -75.64 161.00 97.86 182.77 -121.21 -6.780 75.66

Table 3.3 Estimates of the parameters of AR(k)-GARCH(1,1)L(1) model.

Notes: *, ** and *** indicate significance at 1%, 5% and 10% levels of significance, respectively. MLLV is the

maximized log-likelihood value. Some entries in the table are blank since the estimates of the parameters have

been reported, as expectedly, up to the lag (k) values which have been found by the AIC for the mean models of

the respective countries. Evidently, the value of k has not been found to be the same for all countries.

According to the results reported in this table, the autoregressive coefficients are significant in

varying numbers across the seven countries. The usual GARCH parameters viz., ω, α and β

are statistically significant for all inflation series. Additionally, in agreement with the

Friedman-Ball hypothesis, the estimate of the lagged inflation coefficient θ, is positive and

statistically significant only for two countries viz., Canada and the USA, while for the

remaining countries no significant relationship exists. It is quite possible that this finding of

no such significant effect of inflation on its uncertainty for most viz., five, of the G7 countries

may be due to the fact that this model does not factor for regime-specific inflation behaviour,

which might have masked potentially different realizations due to probable regime shift in

inflation series, especially because the span of the data set is quite large

3.2.2. The Proposed DTGARCH(1,1)L(1) Model

The DTGARCH(1,1)L(1) model, as specified in equations (2.3) and (2.4), incorporates

differential behaviour in both inflation and inflation uncertainty from consideration of regime

switching and also allows for the coefficient attached to the lagged inflation term in inflation

uncertainty model specified in equation (2.4) to be different in the two regimes. This model is

locally i.e., regime-wise linear but overall nonlinear in nature. The estimates12

of the

parameters of this model are presented in Table 3.4.

12 For writing the GAUSS codes of DTGARCH (1,1)L(1) model, we have made use of codes from the GAUSS

programs developed by Philip Hans Franses and Dick Van Dijk on different volatility models, which were

downloaded from http://www.few.eur.nl/few/few/people/frances.

http://www.few.eur.nl/few/few/people/frances


43

Canada France Germany Italy Japan The UK The USA

Conditional mean for the low-inflation regime

ϕl

0 0.007 0.005 0.010 -0.011 -0.022 -0.053 * -0.001

ϕl

1 0.069 0.326 * 0.141 0.261 * -0.020 -0.050 0.394 *

ϕl

2 0.087 -0.033 0.134 ** 0.283 * 0.014 0.328 * 0.025

ϕl

3 0.153 ** 0.029 0.163 ** 0.046 0.022 0.215 * -0.079

ϕl

4 0.078 0.171 ** 0.109 *** -0.026 0.013 -0.084 0.155 **

ϕl

5 0.160 ** -0.092 -0.011 0.018 0.109 *** -0.055 -0.031

ϕl

6 -0.073 -0.031 0.120 ** 0.113 ** -0.009 0.178 * 0.121 ***

ϕl

7 0.000 0.116 0.052 0.186 * 0.100 *** - -0.021

ϕl

8 - 0.057 0.089 *** -0.035 0.009 - -0.106

ϕl

9 - 0.041 - -0.043 0.112 ** - 0.130 **

ϕl

10 - 0.075 - 0.049 0.094 - 0.086

ϕl

11 - - - -0.029 0.013 - -

ϕl

12 - - - -0.152 * -0.084 - -

Conditional mean for the high-inflation regime

ϕh

0 0.018 -0.015 0.014 -0.006 -0.003 -0.018 -0.006

ϕh

1 -0.044 0.401 * 0.034 0.221 ** 0.121 0.425 * 0.325 *

ϕh

2 0.276 * 0.141 *** 0.208 * 0.217 * 0.094 0.263 * 0.020

ϕh

3 0.032 0.169 ** 0.047 0.043 0.097 0.029 0.019

ϕh

4 -0.004 -0.072 -0.003 0.157 * 0.099 -0.031 0.040

ϕh

5 0.055 0.023 0.030 0.089 *** 0.129 *** 0.147 * -0.013

ϕh

6 0.020 0.125 0.114 ** 0.256 * 0.001 0.138 * -0.012

ϕh

7 0.089 0.021 0.005 -0.020 0.076 - 0.146 **

ϕh

8 - 0.042 0.072 -0.034 -0.030 - -0.001

ϕh

9 - -0.047 - 0.137 * 0.144 ** - -0.040

ϕh

10 - -0.007 - 0.139 ** 0.062 - 0.027

ϕh

11 - - - -0.157 * 0.060 - -

ϕh

12 - - - -0.251 * -0.195 * - -

Conditional variance for the low-inflation regime

ωl 0.027

** 0.004 ** 0.011 *** 0.002 0.006 0.002 0.004 ***

αl 0.605

* 0.363 ** 0.604 * 0.255 *** 0.000 0.640 * 0.304 *

βl 0.468

* 0.710 * 0.492 * 0.323 * 0.555 * 0.529 * 0.619 *

θl 0.111

** 0.034 *** 0.046 -0.055 * -0.154 ** -0.017 0.010

Conditional variance for the high-inflation regime

ωh 0.002 0.033

* 0.008 0.000 0.000 0.009 ** 0.000

αh 0.000 0.120 0.000 0.948

* 0.222 ** 0.513 * 0.001

βh 0.658

* 0.001 0.280

* 0.020 0.837

* 0.000 0.655

*

θh 0.119

* -0.039 0.154 * 0.153 * -0.003 0.198 * 0.124 *

MLLV -63.51 172.46 104.54 198.89 -116.26 16.39 88.40

Wald test for

H0: θl = θh 0.02 2.69 5.42 ** 33.89 * 3.51 *** 13.40 * 11.19 *

Table 3.4 Estimates of the parameters of the DTGARCH(1,1)L(1) model.

Notes: *, ** and *** indicate significance at 1%, 5% and 10% levels of significance, respectively. MLLV is the

maximized log-likelihood value. Some entries in the table are blank since the estimates of the parameters have

been reported, as expectedly, up to the lag (k) values which have been found by the AIC for the mean models of

the respective countries. Evidently, the value of k has not been found to be the same for all countries.

Several findings from the estimation results are worth mentioning: First, the estimates of the

parameters in conditional mean clearly establish regime switching behaviour in inflation for

all seven series since some of the own lags in both the regimes are found to be statistically

significant. Second, looking at the parameters of the model for the conditional variance in the


44

two inflation regimes, we first note that in case of Canada, France, Germany, Japan, and the

USA, only one of the parameters of αl and α

h is significant, while for Italy and the UK, both

parameters are significant. Further, only one of the parameters βl and β

h are significant for the

three countries viz., France, Italy, and the UK, and for the remaining ones i.e., Canada,

Germany, Japan, and the USA, both coefficients are significant. Considering two regimes

based on a past level of inflation for the conditional variance which measures inflation

uncertainty, is found to be relevant and statistically meaningful. Regarding the effect of

inflation on inflation uncertainty, i.e., θl and θ

h, we note that both these coefficients are

significant in two countries only, namely, Canada and Italy, while for the remaining five

countries, at least one of θl and θ

h is significant. This establishes the fact that the effect of

inflation on inflation uncertainty is also regime specific. Hence, on the whole, any policy

measure on inflation should be based, inter alia, on regime consideration of inflation.

Finally, the most important hypothesis for the proposed model is whether inflation affects

inflation uncertainty differently in the two inflation regimes. In terms of the parameters, the

null and alternative hypotheses are H0: θ

l =

θ

h and H1:

θ

l ≠

θ

h, respectively. This null

hypothesis has been tested by using the Wald test, and the test statistic values for the seven

series are reported in the last row of Table 3.4. The results of the Wald test show that the null

hypothesis is rejected for all but Canada and France. Regarding the latter two countries, we

can conclude that in the case of Canada, inflation has a positive impact on inflation

uncertainty and does not vary with the regime change while for France, the only conclusion

that can be drawn is that the coefficients do not change significantly between the two regimes.

The Wald test results thus suggest that significant difference in the two regimes exists insofar

the effect of inflation on inflation uncertainty is concerned in case of five members of the G7

countries. These countries are Germany, Italy, Japan, the UK, and the USA.

The positive and significant values of θh in four of these countries viz., Germany, Italy, the

UK, and the USA indicate that inflation increases inflation uncertainty at the high-inflation

regime in each of these countries, while the effect is insignificant for Germany, the UK, and

the USA at the low-inflation regime and negative for Italy. This evidence for Germany, the

UK, and the USA thus supports the findings of Ungar and Zilberfarb (1993) viz., that inflation

affects inflation uncertainty only in the high-inflation regime but not in the low-regime. The

finding that θˆl is negative and significant for Italy and Japan is, however, somewhat unusual

although not exceptional. This means that at the low-regime inflation has a negative effect on

inflation uncertainty or in other words, a reduction in inflation in the low-regime increases

inflation uncertainty, and thus the Friedman-Ball hypothesis fails to hold for these two

countries. Such evidence has been found by a few other studies as well – although with

different volatility specifications. For instance, based on a study on 12 European Monetary

Union countries, Caporale and Kontonikas (2009) have found that during the post-1999

period when average inflation was low, a further reduction in inflation led to an increase

rather than a decrease in inflation uncertainty for a number of countries in the Euro Zone. In a

recent study based on monthly data on US inflation over the period from 1926 to 1992,

Hwang (2001) has found that inflation affects its uncertainty weakly and negatively during

the periods of both high and low inflation.

We report on the Ljung-Box test statistic values based on residuals of this model for all G7

countries in Table 3.5.

The test statistic has been computed for both the standardized and squared standardized

residuals. It is evident from Table 3.5 that none of these are significant, and hence we


45

conclude that the proposed models for both the conditional mean and conditional variance of

inflation along with the chosen lag values are adequate for all G7 countries.

Country Q(1) Q(5) Q(10) Q2(1) Q

2(5) Q

2(10)

Canada 0.007 3.746 7.027 0.670 1.386 4.238

France 0.107 1.975 4.710 0.014 4.348 5.481

Germany 0.369 2.123 4.823 0.638 3.656 7.972

Italy 0.024 2.434 11.420 2.281 5.418 12.963

Japan 0.027 0372 1.480 0.002 2.112 8.114

The UK 0.001 2.676 6.196 0.003 2.303 11.887

The USA 0.014 2.604 5.050 1.501 7.512 9.926

Table 3.5 Results of the Ljung-Box test with standardized residuals and squared standardized residuals.

Notes: *, ** and *** indicate significance at 1%, 5% and 10% levels of significance, respectively. Q(.) and Q2(.)

denote the Ljung-Box test for autocorrelation in standardized residuals and squared standardized residuals,

respectively.

Finally, we make a comparison between the benchmark AR(k)-GARCH(1,1)L(1) model and

the proposed DTGARCH(1,1)L(1) model by the likelihood ratio (LR) test to find if

introduction of regimes based on low and high inflation in both conditional mean and

conditional variance has led to any significant gain in understanding the effects of inflation on

inflation uncertainty. We report the LR test statistic values in Table 3.6.

Country

AR(k)-GARCH(1,1)L(1)

versus

DTGARCH(1,1)L(1)

Canada 24.26 **

France 22.92 ***

Germany 13.36

Italy 32.24 **

Japan 9.90

The U.K. 46.34 * The U.S.A. 25.48 **

Table 3.6 LR test statistic values.

Notes: *, ** and *** indicate significance at 1%, 5% and 10% levels of significance, respectively. Q(.) and Q2(.) denote the Ljung-Box test for autocorrelation in standardized residuals and squared standardized residuals,

respectively.

We observe from this table that the LR test statistic values are significant for five of the G7

countries viz., Canada, France, Italy, the UK, and the USA, and hence we can conclude that

the proposed model explains the relationship ‘better’ than the benchmark model.

4. CONCLUSIONS

The effect of inflation on inflation uncertainty has been studied for the G7 countries in terms

of a model called the DTGARCH(1,1)L(1) model, where the conditional mean as well as the

conditional variance of inflation are based on a consideration of two regimes for inflation, and

the conditional variance specification for each regime is assumed to be GARCH with an

additional term of inflation with a lag of 1. The regimes are determined by the level of

(stationary) inflation of the preceding lag being negative or positive.

The findings on the twin issues of structural break and stationarity of the series, based on the

recently-developed tests of Perron and Yabu (2009) and Kim and Perron (2009), are that all of

the series are trend stationary with a break in their respective trend functions. The estimated

break date thus obtained for each country seems to broadly give empirical support to the


46

phenomenon of ‘great inflation’, which refers to the high and volatile inflation that occurred

in the mid-1960s and lasted for almost twenty years

Regarding the nature of the relationship, the empirical findings clearly show that the impact

of inflation on inflation uncertainty is different in the two regimes for five of the G7 countries

viz., Germany, Italy, Japan, the UK, and the USA. For Canada, the Friedman-Ball hypothesis

for a positive effect of inflation on its uncertainty holds, but this effect is invariant to the two

regimes. Further, the Friedman-Ball hypothesis holds only in the high-inflation regime for

Germany, Italy, the UK, and the USA. On the other hand, the relationship is found to be

insignificant for Germany, the UK, and the USA in the low-inflation regime, while it is

negative and significant for Italy and Japan. This suggests that in the low-inflation regime a

decline in inflation has led to an increase in inflation uncertainty for Italy and Japan, and this

obviously counters the Friedman-Ball hypothesis. Thus, these findings, on one hand, provide

strong empirical support to the Friedman-Ball hypothesis for high-inflation regimes of most

of the G7 countries, and on the other, clearly establish the importance of considering regime-

specific behaviour in modelling the impact of inflation on inflation uncertainty.

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